## 1. Introduction

[2] HF radio signals reflecting from the ionosphere can suffer from substantial multipath delay spread (of the order milliseconds) since they may arrive at the receiver via a number of different routes (e.g., 1-hop F and 2-hop F). As well as causing signal fading, large delay spreads can also reduce the data rates achievable with acceptable bit error rates in digitally modulated systems. A number of authors [e.g.,*Angling et al.*, 1998; *Warrington and Stocker*, 2003; *Stocker and Warrington*, 2011a] have measured the delay spreads found at various latitudes and some of these values have been incorporated into the appropriate ITU-R recommendation [*International Telecommunication Union Radiocommunication Sector* (*ITU-R*), 2000]. Various models have been produced [e.g., *Watterson et al.*, 1970; *Mastrangelo et al.*, 1997; *Angling and Davies*, 1999; *Warrington et al.*, 2006] that allow the effect of the delay spreads on modems to be simulated and tested [e.g., *Angling and Davies*, 1999; *Jodalen et al.*, 2001; *Warrington et al.*, 2011].

[3] There are a number of methods by which the delay spreads on a given propagation path might be predicted. While raytracing (provided the simulated ionosphere is of sufficient accuracy) would form a physically realistic, albeit somewhat computationally intensive, method, the Voice of America Coverage Analysis Program (VOACAP) [*Lane*, 2001] and ITU-R Recommendation 533 [*ITU-R*, 2009] have generally been used for most studies [e.g., *Rogers*, 2003; *Smith and Angling*, 2003]. However, these methods have some limitations, e.g., they only include the effect of specular reflections, only three hops from each ionospheric layer are calculated, and the underlying databases are monthly and hourly in nature, so predictions cannot be made for individual days. The ITU-R have published a method of calculating the delay spread using such prediction techniques [see*ITU-R*, 2007; *Barclay et al.*, 2009] and this has been incorporated in the software package REC533 (now called ITUHFPROP), but not VOACAP, available from the Institute for Telecommunications Services (ITS) website (elbert.its.bldrdoc.gov). However, this implementation is based on using the median signal power or signal-to-noise ratio (SNR), which, as will be demonstrated in this paper, can often lead to incorrectly predicted values. It should also be noted that the multipath probability (MPROB) value given by VOACAP is not the probability of multipath occurring, but is rather the reliability (i.e., the probability that the SNR exceeds a user-defined required SNR) of the modes meeting the multipath criteria set by the user.

[4] A recent paper [*Stocker and Warrington*, 2011a, 2011b] presented a ‘brute force’ method of calculating ITU-R multipath spread (IMPS) [*ITU-R*, 2007; *Barclay et al.*, 2009] from VOACAP predictions that included the effect of the SNR probability density functions (pdf). The predictions of spread were compared with observations obtained over several sub-auroral paths with generally good agreement at sunspot minimum, but poorer performance at sunspot maximum because the presence (or otherwise) of high order modes (e.g., 3F2) is not well predicted by VOACAP at that time for these paths. While the method used to calculate the multipath spread from VOACAP predictions has been reasonably well validated, it suffers from very poor computational efficiency and takes too long to run to be used for practical purposes. For example, if four ionospheric modes are present, it can take up to 20 min to calculate the IMPS for a single frequency-hour-month. This time increases rapidly if additional modes are present. In this paper, a Monte Carlo approach has been adopted that provides a solution that is asymptotically approached by the ‘brute force’ method previously employed, but in run times many orders of magnitude smaller. The matlab code implementing this new method is available from the author and will allow the users of prediction programs such as VOACAP to properly determine the expected multipath spreads and the likely effect that these will have on reception.